Performance Of MIMO Channel Over SISO

Performance Of MIMO Channel Over SISO

S. Venkatesh*,D .Chiranjeevi*, A. Rama Krishna**

*FINAL YEAR B.TECH, ECE, K L UNIVERSITY, Vaddeswaram, AP, India

**Associate Professor B.Tech, Dept. Of ECM, K L University, Vaddeswaram, AP, India

Abstract

During the last decade, the demand for capacity in wireless local area networks and cellular mobile systems has grown in a literally explosive manner. In particular, compared to the data rates made available by todays technology, the need for wireless Internet access and multimedia applications require an increase in information throughout with order of magnitude. One major technological breakthrough that will make this increase in data rate possible is the use of multiple antennas at the transmitters and receivers in the system. In this paper, a tutorial introduction on the channel capacity of a MIMO channel will be given.

1. Introduction

   Due to the ever increasing demand of faster data transmission speed in the recent or future telecommunication systems. In conventional wireless communications, a single antenna is used at the source,

   and another single antenna is used at the destination. In some cases, this gives rise to problems with multipath

   different input sequences may give rise to the same output sequence, causing different input sequences to be confusable at the output. To avoid this situation, a non-confusable subset of input sequences must be chosen so that with a high probability, there is only one input sequence causing a particular output. It is then possible to reconstruct all the input sequences at the output with negligible probability of error. A measure of how much information that can be transmitted and received with a negligible probability of error is called the channel capacity. To determine this measure of channel potential, assume that a channel encoder receives a source symbol every Tssecond. With an optimal source code, the average code length of all source symbols is equal to the entropy rate of the source. If S represents the set of all source symbols and the entropy rate of the source is written as H(S), the channel encoder will receive on averageH(S)/T(S) informationbits per second.1 Assume that a channel codeword leaves the channel encoder every Tcsecond. In order to be able to transmit all the information from the source, there must be

   effects. When an electromagnetic field (EM field) is met with obstructions such as hills, canyons, buildings,

   R = ()

   (1)

   and utility wires, the wave fronts are scattered, and thus they take many paths to reach the destination. The late arrival of scattered portions of the signal causes problems such as fading, cut-out (cliff effect), and intermittent reception (picket fencing). In digital communications systems such as wireless Internet, it can cause a reduction in data speed and an increase in the number of errors. The use of two or more antennas, along with the transmission of multiple signals (one for each antenna) at the source and the destination, eliminates the trouble caused by multipath wave propagation, and can even take advantage of this effect.

2. CHANNEL CAPACITY

   At the input of a communication system, discretesource symbols are mapped into a sequence of channel symbols. The channel symbols are then transmitted/ conveyed through a wireless channel that by nature is random. In addition, random noise is added to the channel symbols. In general, it is possible that two

   information bits per channel symbol. The number R is

   called the information rate of the channel encoder.The maximum information rate that can be used causing negligible probability of errors at the output is called the capacity of the channel. By transmittinginformation with rate R, the channel is used everyTcseconds. The channel capacity is then measuredin bits per channel use. Assuming that the channelhas bandwidth W, the input and output can be representedby samples takenTs

   =1/2W seconds apart.With a band-limited channel, the capacity is measured in information bits per second. It is common to represent the channel capacity within a unit bandwithof the channel. The channel capacity is then measured in bits/s/Hz.

   It is desirable to design transmission schemes thatexploit the channel capacity as much as possible. Representing the input and output of a memory less wireless channel with the random variables X and Y respectively, the channel capacity is defined as below.

   C = max I (x;y), (2)

   whereI(X;Y) represents the mutual information between X and Y . Eq.(2) states that the mutual information is maximized with respect to all possible transmitter statistical distributions p(x). Mutual information is a measure of the amount of information that one random variable contains about another variable. The mutual information between X and Y canalso be written as

   I (X;Y)=H(Y)-H(Y/X), (3)

   whereH(Y |X) represents the conditional entropy betweenthe random variables X and Y. The entropy of a random variable can be described as a measure of the amount of information required on average to describe the random variable. It can also described as a measure of the uncertainty of the random variable. Due to (3), mutual information can be described as the reduction in the uncertainty of one random variable due to the knowledge of the other. Note that the mutual information between X and Y depends on the properties of the channel (through a channel matrix H) and the properties of X (through the probability distribution of X). The channel matrix H used in the representation of the input/output relations of a MIMO channel is defined in the next section.

3. MIMO System Model

   Consider a Multi-Input communication system consisting with Nttransmit antennas, Nr receiver antennas as shown in the figure 1,a narrowband time- invariant wireless channel can be represented by Nt X Nr matrix. Consider a transmitted symbol vector x, which is composed of Nt independent input symbols x1,x2.xNt. Then received signal y can be written in a matrix form as follows:

   H

   H

   Ex

   Nt X+Z(4)

   Where Z=(z1,z2,zr)r which is assumed to be zero-mean circular symmetric complex Gaussian. ZMCSCG noise with covariance matrix, E{nnH}=N0INr. The autocorrelation of transmitted signal vector is defined as Rxx=E{xxH} the transmission power for each antenna is assumed to be 1. Then Tr(Rxx)=Nt. A general entry of the channel is denoted by {hij}. This represents the complex gain of the channel between the jth transmitter and ithreceiver . With a MIMO system consisting of Nt transmitter and Nr receiver antennas, the channel matrix is written as

   11

   = (5) 1

4. SISO CHANNEL CAPACITY

   The ergodic (mean) capacity of a random channel with nT= nR = 1 and an average transmit power constraint PT can be expressed as [2]

   C = EH{ max I(X;Y)}, (6)

   Where P is the average power of a single channel codeword transmitted over the channel and EH denotes the expectation over all channel realizations. Compared to the definition in (2), the capacity of the channel is now defined as the maximum of the mutual information between the input and the output over all statistical distributions on the input that satisfy the power constraint. If each channel symbol at the transmitter is denoted by s, the average power constraint can be expressed as

   P = E [|s|2] PT (7)

   Fig 1. MIMO Communication system Model

5. Deterministic MIMO channel capacity:

   The maximum information rate that can used causing negliible probability of errors at the output is called

   = E{HxxHHH+zzH}

   the capacity of the channel. The capacity of a

   H H H

   deterministic channel is defined as

   C = max I(x;y) bits/channel use (8)

   Where I(x;y) is mutual information of random vector x and y. Eq.(8) states that mutual information is maximized with respect to all possible transmitter statistical distribution f(x). Mutual information is a measure of the amount of information that one random variable contains about another variable. The mutual information of the two continues random vector ,x and y, is given as

   I(x;y)=H(y)-H(y/x) (9)

   In which H(y) is the differential entropy of y and H(y/x) is the conditional differential entropy of y when

   =HE{xx }H +E{zz } (11)

   Where Ex is the energy of the transmitted signals, and N0 is the power spectral density of the additive noise. The differential entropy H(y) is maximized when y is ZMCSCG, which consequently requires x to be ZMCSCG as well. Then , the mutual information of y and z is respectively given as

   H(y)=log2{det{eRyy}} H(y)=log2{det{eNoINr}} (12)

   Using Equation (12), the mutual information of Equation (10) is expressed as

   x is given. Using the statistical independence of two random vectors Z and X in equation (4), we can show

   I(x;y)= log2det (INr +

   RxxHHH) bps/Hz

   the following relationship:

   C= max log det (I

   + R

   HHH) bps/Hz (13)

   H(y/x)=H(z)

   2 Nr

   xx

   I(x;y)=H(y)-H(z) (10)

   From equation (10), given than H(z) is a constant, we can see that the mutual information is maximized when H(y) is maximizes. Note that the mutual information between x and y depends on the properties of the channel (through channel matrix) and properties of x (through the probability distribution of x).

   Using Equation (5), meanwhile, the auto correlation matrix of y is given as

6. Channel knowledge at receiver:

   When the transmitter has no knowledge about the transmitter, it is optimal to spread the energy equally among all transmitter antennas. That is the autocorrelation function of the transmitter signal vector x is given as

   Rxx=INt (14)

   In this case channel capacity is given as

   C=log2det (INr

   +

   HHH) (15)

   R = E{yyH} = + +

   Using eigen value decomposition the matrix product is

   yy

   written as

   HHH=QQH (16)

   =E +

   Where Q is the eigen vector matrix with orthogonal

   columns and ^is a diagonal matrix with the eigen values on the main diagonal.

   SVD of H yields H=uVH, inserting this into Equation

   (17) and rearranging the parameters we get,

   ~ H

   H ~ H

   C=log2det (INr

   +

   QQ)=log2det (INr

   + )

   =( U V V+U z)

   ~ ~ H

   =log2det (INr + i)

   It is easier to see that the total capacity of MIMO channel is made up by the sum of parallel AWGN channels. The number of sub channels is determined by the rank of the channel matrix where r denotes the rank matrix , that is r= Nmin min(Nt,Nr). When CSI is not available at the transmitter and thus, the total power is equally allocated to all transmit antennas.

7. Channel Knowledge available at the transmitter side:

   When the channel state information available at the transmitter, modal decomposition can be performed as shown in the figure (8), in which a transmitted signal is

   preprocessed with V in the transmitter and then, a

   =( +U z) (18)

   Which is equivalent to the following r virtual SISO Channel, that is

   ~

   =( i+ z, i=1,2.r. (19)

   If the transmit power for the ith transmit antenna is given by rtthe capacity of the i th virtual SISO channel is

   i i 2 i

   i i 2 i

   C ( )= log (1+ ), i=1,2.r (20)

   Assume that total available power at the transmitter is limited to

   =0

   =0

   received signal is post processed with UH in the receiver.

   E{xxH} =

   { 2} = Nt (21)

   Where the total power constraint in Equation (21) must be satisfied. The capacity in equation (21) can be maximized by solving the following allocation problem:

   C=max

   (1+ ) (22)

   Figure 3. System model when channel knowledge is available at the transmitter side

   =1 2

   i=1 i t

   i=1 i t

   Subjected to r = N .

   i

   ~

   H

   H

   =U ( Hx+z)

   ~ ~

   The capacity can be increased by resorting to the so-

   called water filling principle, by assigning various levels of transmitted power to various transmitting antennas. This power is assigned on the basis that the better the channel gets, the more power it gets and vice

   =(

   UHHV+UHz) (17)

   versa. This is an optimal energy allocation algorithm.

8. Simulation results:

   It is found that, assuming perfect channel knowledge at the receiver side, capacity increases linearly with the min(Nt , Nr).

   Figure 4.1 MIMO Channel capacity for SISO,SIMO, MISO and MIMO when CSI not available at the transmitter.

   Figure 4.2 Channel capacity comparisons with out and with CSI at transmitter.

   Transmitter side channel knowledge has good affect on the capacity if SNR is low. However, if SNR is high channel knowledge at the transmitter side does not have a noticeable effect in the capacity increase. When channel distribution knowledge is available at both

   sides, increasing the number of antennas in highSNR have only a negligible effect in capacity increase, at moderate SNR the increase in capacity is greater, but now very effective.

   Antenna correlation also plays a role in channel capacity. If the SNR is low, and assuming channel knowledge both side, antenna correlation increases capacity, however it decreases channel capacity at high SNR.

   Conclusion

   This paper describes the capacity calculation of MIMO system. We have presented simulation results comparing channel capacities of SISO, SIMO, MISO and MIMO formula. The simulation results shows that MIMO system with CSI available at the transmitter can greatly improve spectral efficiency over MIMO system without CSI at transmitter.

9. REFERENCES

1. S. Catreux, P. F. Driessen, L. J.Greenstein, Attainable throughput of an interference-limited multiple-input multiple-output (MIMO) cellular system, IEEE Transactions on Communications, 49(8):13071311, aug 2001.

2. T. M. Cover, J. A. Thomas, Elements of Information theory, John Wiley & Sons, Inc., 1991.

3. I. Telatar, Capacity of multi-antenna gaussian channels, AT&T Technical Memorandum, jun 1995.

4. G. J. Foschini, M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless PersonalCommunications, 6:311335, aug 1998.

5. J. G. Proakis, Digital Communications, McGraw-Hill, Inc., 1995.

AUTHORS

First author and corresponding author:

SOLASA VENKATESH was born in Guntur. He is now pursuing B.Tech Degree in Electronics And Communication engineering from K L University of Guntur. He is interested in Communications and Networking.

E-mail: solasa.venkatesh@yahoo.com

DUPATI CHIRANJEEVIwas born in Guntur. He is now pursuing B.Tech Degree in Electronics And Communication engineering from K L University of Guntur. He is interested in Communications and Networking.

E-mail: dupati.chiru802@gmail.com

AKELLA RAMAKRISHNA is working as associate professor in K L UNIVERSITY. He is interested in Intelligent systems and networks.

E-mail: ramakrishna.a@kluniversity.in